# Properties

 Label 1.27.ak Base Field $\F_{3^{3}}$ Dimension $1$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{3^{3}}$ Dimension: $1$ L-polynomial: $1 - 10 x + 27 x^{2}$ Frobenius angles: $\pm0.0877398280459$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-2})$$ Galois group: $C_2$ Jacobians: 1

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $1$ Slopes: $[0, 1]$

## Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 18 684 19494 530784 14347458 387423756 10460425014 282430166400 7625601845298 205891158689964

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 18 684 19494 530784 14347458 387423756 10460425014 282430166400 7625601845298 205891158689964

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{3}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-2})$$.
All geometric endomorphisms are defined over $\F_{3^{3}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{3}}$.

 Subfield Primitive Model $\F_{3}$ 1.3.c

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.27.k $2$ 1.729.abu